Date & Time: October 28, 2015, 15:00-16:00.
Venue: Ramanujan Hall
Title: Invariants of several matrices under SL(n) \times SL(n)-action.
Speaker: K. V. Subrahmanyam, CMI Chennai
Abstract: Let $R(m,n)$ denote the ring of invariant polynomial functions of the $ SL(n) \times SL(n) action $ on $m$ tuples of matrices. We describe the ring of relations (the second fundamental theorem) among these invariants. We also describe the $S(m) \times S(m)$-module structure of the invariant ring, where $S(m)$ denotes the symmetric group on [m], and also the module structure for the diagonal action of $S(m)$. We describe another natural relation among the invariants which we believe will be useful to give an upper bound on the degree in which the ring of invariants is generated. We give an algorithm to detect when a tuple of matrices is in the null cone of this action, which runs in time polynomial in the degree in which the invariant ring is generated. This algorithm rides on an algorithm of Gurvits, and our analysis is based on a novel use of blow-ups of matrices, which we will outline. We will also state some recent developments, a polynomial time algorithm for the same problem by two independent groups of researchers. This is joint work with Gabor Ivanyos and Youming Qiao.